Question

Group or not group? The set of M_nxn (R) of all nxn matrices under multiplication.

Answer 1

Question. Group or not group? The set of M_nxn (R) of all nxn matrices under multiplication.

Answer. This set is not a group under multiplication.

Proof. Let $A\in M_{n\times n}(R)$, $A=\begin{pmatrix} 0 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & 0 \end{pmatrix}$.

Suppose that matrix $A$ is invertible (there exists an $B\in M_{n\times n}(R)$, such that $A\cdot B=B\cdot A=I_n$).

On the another hand, for every matrix $C\in M_{n\times n}(R)$ we have $A\cdot C=C\cdot A=0$.

Therefore, in the set of all matrices $n\times n$ there exists a non-invertible matrix, so $M_{n\times n}(R)$ isn't a group.